Question

Question: Suppose $f\left(x\right),g\left(x\right)\in R\left[x\right]$ and $f\left(x\right)\equiv g\left(x\right)\left(\text{modx}\right)$. What can be said about the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$?

Short answer

The graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ intersect the y-axis.

Step-by-step solution

 Step 1: Given part

It is given that $f\left(x\right),g\left(x\right)\in R\left[x\right]$ and $f\left(x\right)\equiv g\left(x\right)\left(\text{mod}x\right)$.

Proof

From exercise 9, it is known that $f\left(x\right)\equiv g\left(x\right)\left(\text{mod}p\left(x\right)\right)$;if and only if $f\left(x\right)$ and$g\left(x\right)$leave the same remainder when divided by $p\left(x\right)$.

From exercise 9, for $f\left(x\right)$ and$g\left(x\right)$,there will be a constant term,which areequal. So, $f\left(\mathbf{0}\right)=g\left(\mathbf{0}\right)$.

Therefore, the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ intersect the y-axis.